Testing electron–phonon coupling for the superconductivity in kagome metal CsV3Sb5

In crystalline materials, electron-phonon coupling (EPC) is a ubiquitous many-body interaction that drives conventional Bardeen-Cooper-Schrieffer superconductivity. Recently, in a new kagome metal CsV3Sb5, superconductivity that possibly intertwines with time-reversal and spatial symmetry-breaking orders is observed. Density functional theory calculations predicted weak EPC strength, λ, supporting an unconventional pairing mechanism in CsV3Sb5. However, experimental determination of λ is still missing, hindering a microscopic understanding of the intertwined ground state of CsV3Sb5. Here, using 7-eV laser-based angle-resolved photoemission spectroscopy and Eliashberg function analysis, we determine an intermediate λ=0.45–0.6 at T = 6 K for both Sb 5p and V 3d electronic bands, which can support a conventional superconducting transition temperature on the same magnitude of experimental value in CsV3Sb5. Remarkably, the EPC on the V 3d-band enhances to λ~0.75 as the superconducting transition temperature elevated to 4.4 K in Cs(V0.93Nb0.07)3Sb5. Our results provide an important clue to understand the pairing mechanism in the kagome superconductor CsV3Sb5.

experimental estimation of orbital-and momentum-dependent λ and its possible connection with superconductivity are highly desired to understand the nature of the superconductivity in CsV 3 Sb 5 . Here we experimentally extract the orbital-and momentum-dependent λ p,d (k) by determining the EPC-induced kinks in the electronic band structure. Our results reveal an intermediate EPC with λ=0.45-0.6 in CsV 3 Sb 5 , which can support a T c on the same magnitude of the experimental value. Intriguingly, we find that λ d is enhanced by about 50% in the isovalent-substituted Cs(V 0.93 Nb 0.07 ) 3 Sb 5 with an elevated T c = 4.4 K. Our results suggest that EPC can play an important role on the superconductivity in CsV 3 Sb 5 .
Results Figure 1a, c shows the crystal structure and Fermi surface (FS) topology of CsV 3 Sb 5 , respectively. In agreement with previous DFT and ARPES studies 12,[19][20][21] , the Sb 5p-band forms a circular FS, marked as α, at the BZ center and the V 3d bands yield hexagonal and triangle FSs, marked as β and δ in Fig. 1c, respectively. Figure 1d shows a typical ARPES intensity plot of the α band corresponding to the black cut shown in Fig. 1c. The coupling between electrons and bosonic modes is manifested by the intensity and dispersion anomalies, known as kink 23,24 , near a binding energy E B~3 2 meV. This many-body effect can be quantified by fitting the ARPES momentum distribution curves (MDCs) with a Lorentzian function 25 : Iðk, ωÞ / Aðk, ωÞ = 1 π ImΣðωÞ ðω À εðkÞ À ReΣðωÞÞ 2 + ImΣðωÞ 2 , ð1Þ where ReΣ(ω = E-E F ) and ImΣ(ω = E-E F ) are the real and imaginary parts of the single-particle self-energy. ε(k) is the non-interacting bare band that can be approximated as a liner dispersion crossing E F 23 . Figure 1e demonstrates the extracted self-energy of the α band. We subtract a linear bare band from the experimentally extracted band to obtain ReΣ(ω) (see supplementary note 1). To extract the electron-boson coupling induced ImΣ(ω), the electron-electron and electron-impurity scatterings induced self-energy effects are removed, as suggested by previous practices 23,26 (see supplementary note 2). At E B~3 2 meV, a peak near in ReΣ(ω) and a step jump in ImΣ(ω) prove strong manybody interactions. Since the self-energy anomalies persist above CDW transition temperature T CDW (supplementary Fig. S7), we attribute the self-energy anomaly to EPC. Figure 2a-c compares the EPC-induced kinks on the α and β bands at 6 K. The ARPES intensity plots of the α and β bands shown in Fig. 2b correspond to the black cuts in Fig. 2a. While the kink near E B~3 2 meV is clear on both the α and β bands, an additional kink is observed at a lower E B~1 2 meV on the β band (Fig. 2c). The 12-meV kink is also Temperature Pressure or chemical doping  prominent in ReΣ(ω). As we show in Fig. 2d, ReΣ(ω) of the β band shows a peak near E B~1 2 meV, proving strong d-electron-phonon coupling near 12 meV. In contrast, ReΣ(ω) of the α band only shows a broad shoulder.
The observation of clear EPC effects on both 5p and 3d bands points to a non-neglectable role of EPC for superconductivity in CsV 3 Sb 5 . To test the EPC-driven superconductivity, we extract the Eliashberg function, α 2 F ω ð Þ, at T = 6 K, slightly above T c , using the maximum entropy method 27,28 (see methods). A fit of the ReΣ(ω) and the extracted α 2 F ω ð Þ are shown in Fig. 2d, e, respectively. λ and the logarithmic mean phonon frequency are obtained via 28,29 : lnω log = 2=λ where ω max is the maximum frequency of the phonon spectrum. As shown in Fig. 2e, the orbital dependence of the EPC is mirrored in the different shapes of α 2 F(ω), where phonon modes near 32 meV are accounted for 70% of the total EPC strength on the α band, λ p , but less than 50% for the EPC strength on the β band, λ d . Interestingly, due to the spectral weight redistribution in α 2 F ω ð Þ (shaded area in Fig. 2e), the extracted λ p and λ d are similar with λ p,d~0 .45 ± 0.05. We also employed the MEM fits the extracted ImΣ(ω), which yields a λ cοnsistent with the ReΣ(ω) fits (see supplementary note 4 and Fig. S2). Theoretically, λ can approximately be derived from a simpler approach 29 following λ dev = À∂ReΣðωÞ=∂ω| ω = E F ffi λ, when T is far lower than the Debye temperature. At T = 6 K, this method yields a λ dev~0 .6±0.1, qualitatively consistent with Eq. (2) within the experimental uncertainty (Fig. 3f).
Generally, EPC can exhibit momentum dependence. Figure 3), we derive T c in a range from 0.8 K to 3 K (see supplementary note 6). The upper limit is comparable to the experimentally determined T c in CsV 3 Sb 5 (Fig. 4a). We shall note that the CDW gap near the M point 20,31 (supplementary Fig. S6f) flattens the δ bands near E F , hindering the precise estimation of EPC strength. However, strong self-energy anomalies are observed on the δ bands and they have the same energy scales as the α and β bands (supplementary Fig. S6).
As shown in Figs. 4a and 1a, T c of CsV 3 Sb 5 is increased with chemical substitutions or external pressure [32][33][34][35] . We thus continue to examine the EPC in a 7% Nb-doped Cs(V 0.93 Nb 0.07 ) 3   ReΣ(ω) of the pristine CsV 3 Sb 5 . Remarkably, we observe that while ReΣ(ω) on the α band is similar in Cs(V 0.93 Nb 0.07 ) 3 Sb 5 and CsV 3 Sb 5 , on the β band, it shows a strong enhancement in the Nb-doped sample, especially near E B~1 0 meV. Based on the extracted α 2 F(ω), shown in Fig. 4e, we find that λ d~0 .75 ± 0.05 is enhanced by about 50% in Cs(V 0.93 Nb 0.07 ) 3 Sb 5 (Fig. 4f). Such giant enhancement is also manifested by the slope of ReΣ(ω) near E F (Fig. 4d). Consequently, the enhanced λ d in Cs(V 0.93 Nb 0.07 ) 3 Sb 5 is expected to elevate T c up to 4.5 K (see supplementary note 6), which is comparable to the experimental value of 4.4 K (Fig. 4a). Such synchronous enhancements of λ d and T c may indicate that the V 3d-electron-phonon couplings are the main driver of the superconductivity in CsV 3 Sb 5 .
Finally, we discuss the influences of CDW order on the quantitative extraction of λ at T < T CDW . The formation of a CDW gap will modify the bare band to deviate from a linear dispersion near E F . As we show in the supplementary Fig. S5, within the experimental resolution, we do not observe a CDW gap on the α and β bands. Therefore, for the α and β bands, the CDW modified bare band dispersion below T CDW is ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ε 2 0 k ð Þ+ Δ 2 CDW q ffi ε 0 ðkÞ, where ε 0 ðkÞ = v 0 _k is the linear bare band dispersion above T CDW . In this case, the linear bare band assumption used in our study is a good approximation. Indeed, the excellent agreement of ReΣ(ω) and ImΣ(ω) linked by Kramers-Kronig transformation 23,26 validates the linear bare band assumption for the α and β bands (supplementary Figs. S1c-d). The linear bare band assumption, however, does not apply to the δ band that forms a CDW gap comparable to the kink energy 20,31 . We also note that the formation of CDW will also modify the electronic self-energy. As we show in the supplementary Fig. S7e, λ dev shows an inflection point at T CDW , which may suggest an enhanced EPC strength below T CDW . However, it can also be a consequence of the CDW-corrected electronic self-energy effect (see supplementary note 8).
In summary, by investigating the electronic kinks, we determined an intermediate EPC that is twice larger than the DFT calculated value in the kagome superconductor CsV 3 Sb 5 and Cs(V 0.93 Nb 0.07 ) 3 Sb 5 . Our results provide an important clue to understand the pairing mechanism in CsV 3 Sb 5 . The orbital, momentum of electronic kinks and their strengthening with the promoted T c prove that the EPC in CsV 3 Sb 5 is strong enough to support a T c comparable to the experiment value and hence cannot be excluded as a possible pairing mechanism. While the exact microscopic pairing mechanism calls for further scrutiny, it is important to point out that the EPC-driven superconductivity is not incompatible with the recently observed pair-density wave (PDW) in CsV 3 Sb 5 17 . Indeed, PDW has been observed in another conventional superconductor NbSe 2 , where the pair-density modulation is due to the real space charge density modulations 36 . We also note that the EPCdriven superconductivity can coexist with the time-reversal symmetrybreaking (TRSB) orders or fluctuations 18,37,38 , as proposed by theoretical studies [39][40][41] . In those cases, the superconducting order parameter is expected to intertwine with the TRSB order parameter, which gives rise to an unconventional ground state.

Growth and characterization of single crystals
Single crystals of CsV 3 Sb 5 were grown using CsSb 2 alloy and Sb as flux. Cs, V, Sb elements and CsSb 2 precursor were sealed in a Ta crucible in a

Laser-ARPES measurements
ARPES measurements were performed for the freshly cleaving surface with a Scienta-Omicron R4000 hemispherical analyzer with an ultraviolet laser (hν = 6.994 eV) at the Institute for Solid State Physics, the University of Tokyo 42 . The energy resolution was set to be 1.3 meV. The sample temperature was set to be 6 K if there is no special announcement. The samples were cleaved in situ and kept under a vacuum better than 3 × 10 −11 torr during the experiments.

Maximum entropy method
The Eliashberg function α 2 F ω; ϵ, k ð Þis related to the real part of the self-energy by the integration function ReΣ ϵ, k; T ð Þ= where K y, y 0 ð Þ= R 1 À1 dx f xÀy ð Þ 2y 0 x 2 Ày 2 and f x ð Þ is the Fermi distribution function. It is an ill-posed problem to obtain the Eliashberg function from Eq. (4). In this work, we adopted the maximum entropy method (MEM) 27,28 , which is frequently used to perform the analytic continuation 43 . By considering the energy resolution of the laser-ARPES, we estimated that the error bar of the real part of the selfenergy was 1 meV. MEM requires a model default function to define the entropic prior. Here, we adopted the following model: where m 0 = 15 meV, ω D = 10 meV, and ω m = 80 meV. This default model was also used in the previous study of the electron-phonon coupling on the Be surface 28 .

Data availability
Data are available from the corresponding author upon reasonable request.

Code availability
Codes are available from the corresponding author upon reasonable request.